CURRENT STATE OF RESEARCH ON THE IONOSPHERE

Ya. L. Al’pert

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
2. General picture of radio-wave propagation in the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . 263
3. Structure of the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
4. Diurnal and seasonal course of the ionosphere. Data on the various layers . . . . . . . . . . . . . . . 278
5. Irregular phenomena in the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
   Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

1. INTRODUCTION

A little more than two decades have passed since the time when, with the aid of radio-engineering methods, the existence of ionized conducting layers above the earth’s surface, situated considerably higher than the stratosphere, was first experimentally demonstrated.^1,2 The region of the atmosphere containing these layers subsequently received the name ionosphere.

During these twenty years the problem of ionospheric research has developed very rapidly, and at the present time it presents a fairly coherent picture. At the same time, a number of fundamental questions remain unresolved, and therefore, despite the known regularity of the picture, it still has many substantial gaps.

In reviewing the journal literature, one can easily see in what direction both experimental and theoretical investigations of the ionosphere have been developing in recent years, and can rather accurately outline the range of unresolved problems. Since in the further exposition we shall touch on these questions only in passing (and not all of them), it may therefore be useful at the very outset to point out some of them.

First of all, the question of the formation of several layers in the ionosphere remains unclear. To some extent this is explained by the fact that the gaseous composition of those regions of the atmosphere in which these layers are located is still not known precisely. There are still few data on the processes that determine the overall ionization balance in each of the layers. It is easy to see how the center of gravity of physical theories of

in the ionosphere has shifted to the study of microprocesses, the study of the ionosphere as a plasma as a whole, the study of the mechanism of ionization, recombination processes, collisions, diffusion, etc.3–12, whereas until quite recently the main part of the work was devoted to the macrotheory of the ionosphere—the problem of the propagation of radio waves in an inhomogeneous anisotropic medium of the ionospheric type.

Secondly, a large place in these investigations is occupied by the problem of the \(F_2\) layer, the question of its morphology. This includes questions concerning the climate of the \(F_2\) layer, the character and causes of its latitude effect, the compression and expansion of the layer, the part of the solar spectrum that causes ionization of the layer, etc.13–20.

Thirdly, phenomena occurring during ionospheric storms have not received a complete explanation.

Fourthly, we shall also point to the uncertainties that arise in analyzing the causes producing the appearance of the sporadic \(E\) layer and a number of other irregular phenomena in the ionosphere.

Within the scope of the present article it does not seem advisable to dwell in detail on questions of the theory of the ionosphere, all the more so since other recently published works are devoted to them21–23; the main attention is given only to the general state of our knowledge of the ionosphere, obtained mainly by radiophysical methods.

2. GENERAL PICTURE OF THE PROPAGATION OF RADIO WAVES IN THE IONOSPHERE

In view of the exceptional role played, on the one hand, by radio methods in investigations of the ionosphere and, on the other hand, by the ionosphere in questions of radio communication, it is advisable to dwell first of all on a consideration of the general results of the theory of the propagation of radio waves in the ionosphere.

a) Refractive index of the ionosphere

State of polarization of the wave

For the analysis of the electrical properties of the medium and of the properties of the wave propagating in it, let us consider the propagation of a plane monochromatic wave

\[ E \sim e^{i\omega\left(t-\frac{\mathbf{N}\mathbf{r}}{c}n\right)} \tag{2.1} \]

in an ionized medium consisting of free electrons. From what follows it will be seen that at the present time it may be considered experimentally proven that the principal role in all layers of the ionosphere is played by electrons, and not by ions.

In formula (2.1) \(\mathbf{N}\) is the normal to the wave front, \(\omega\) is the angular frequency, \(t\) is time, \(\mathbf{r}\) is the radius vector of the point in space, \(c\) is the velocity of light in vacuum, and \(n\) is the refractive index.

Applying the equation of motion of an electron under the action of a wave incident on the layer (taking into account the presence of the external magnetic field of the earth \( \mathbf{H}_0 \)),

\[ m \ddot{\mathbf{r}}_0 = - e\mathbf{E} - \frac{e}{c}[\mathbf{r}\mathbf{H}_0], \tag{2,2} \]

and the relation between the vector of electric induction \((\mathbf{D})\) and the vector of polarization per unit volume of the medium \((\mathbf{P} = -eN_0\mathbf{r}_0)\),

\[ \mathbf{D}=\mathbf{E}+4\pi\mathbf{P}, \tag{2,3} \]

we obtain that the dielectric constant is equal to

\[ \varepsilon' = \left| \begin{array}{ccc} 1-\dfrac{1-h_x^2}{1-h^2}\,v, & \dfrac{h_x h_y - ih_z}{1-h^2}\,v, & \dfrac{h_x h_z + ih_y}{1-h^2}\,v \\[1.2em] \dfrac{h_x h_y + ih_z}{1-h^2}\,v, & 1-\dfrac{1-h_y^2}{1-h^2}\,v, & \dfrac{h_y h_z - ih_x}{1-h^2}\,v \\[1.2em] \dfrac{h_x h_z - ih_y}{1-h^2}\,v, & \dfrac{h_y h_z + ih_x}{1-h^2}\,v, & 1-\dfrac{1-h_z^2}{1-h^2}\,v \end{array} \right|, \tag{2,4} \]

where

\[ \begin{gathered} v=\frac{4\pi N_0 e^2}{m\omega^2}, \qquad h=-\frac{eH_0}{mc\omega}, \\[0.4em] h_x=h\alpha_H,\qquad h_y=h\beta_H,\qquad h_z=h\gamma_H, \end{gathered} \tag{2,5} \]

and \(N_0\) is the number of electrons in \(1\ \text{cm}^3\), \(m\) and \(e\) are, respectively, the mass and charge of the electron, \(H_0\) is the intensity of the earth’s magnetic field, and \(\alpha_H,\ \beta_H,\ \gamma_H\) are the cosines of the angles of the vector \(\mathbf{H}_0\) with the \(x\), \(y\), and \(z\) axes, respectively.*)

We see that the dielectric constant of the ionosphere is a tensor, i.e. we are dealing with an anisotropic medium, namely with the case of magnetoactive artificial anisotropy caused by the external magnetic field of the earth. Naturally, one should thus expect double refraction to be observed here and, in particular, the Faraday effect in the case when \(\mathbf{H}_0\) and \(\mathbf{N}\) are collinear \((\mathbf{H}_0 \parallel \mathbf{N})\), and the Cotton–Mouton effect when \(\mathbf{H}_0\) and \(\mathbf{N}\) are mutually perpendicular \((\mathbf{H}_0 \perp \mathbf{N})\).

Substituting into the wave equation

\[ \Delta \mathbf{E} - \operatorname{grad}\operatorname{div}\mathbf{E} + \frac{\omega^2}{c^2}\mathbf{D}=0 \]

*) In deriving (2,4) we neglect absorption in the medium. This means that the frequency \(\nu\) of electron collisions with neighboring particles is assumed small in comparison with \(\omega\) \((\nu \ll \omega)\).

value of the electric induction vector D, calculated with the aid of (2,3) and (2,4), one can obtain, after some transformations, equations relating the components of the electric field:

\[ \left. \begin{aligned} &E_x(n^2-\varepsilon-n^2\alpha^2)+E_y(-n^2\alpha\beta+i\varepsilon_1)+E_z(-n^2\alpha\gamma)=0,\\ &E_x(-n^2\alpha\beta-i\varepsilon_1)+E_y(n^2-\varepsilon-n^2\beta^2)+E_z(-n^2\beta\gamma)=0,\\ &E_x(-n^2\alpha\gamma)+E_y(-n^2\beta\gamma)+E_z(n^2-\varepsilon_0-n^2\gamma^2)=0, \end{aligned} \right\} \tag{2,6} \]

where

\[ \varepsilon=1-\frac{v}{1-h^2},\qquad \varepsilon_1=\frac{hv}{1-h^2},\qquad \varepsilon_0=1-v \]

and \(\alpha\), \(\beta\), and \(\gamma\) are, respectively, the direction cosines of the normal N to the wave front. To simplify the formulas, the \(Z\)-axis has been chosen to coincide with H\(_0\).

From (2,6) we obtain directly, for the refractive index, the formula

\[ n^2=1-\frac{2v(1-v)} {2(1-v)-h^2(1-\gamma^2)\pm\sqrt{h^4(1-\gamma^2)^2+4h^2\gamma^2(1-v)^2}}, \tag{2,7} \]

in which \(\gamma\) is equal to the cosine of the angle \(\gamma_0\) between the normal N to the wave front and the vector of the Earth’s magnetic field H\(_0\).

From formula (2,7) follows the double refraction of the ionosphere. It is customary to say that the upper sign of the radical in expression (2,4) determines the refractive index \(n_1\) of the so-called ordinary wave, and the lower sign—\(n_2\), of the extraordinary wave, into which the wave propagating in the ionosphere is split. It is easy to see from (2,7) that both waves in the ionosphere are, in essence, extraordinary, since neither of them is spherical. In Fig. 1 the dependence of the quantities \((1:n_1)\) and \((1:n_2)\) on \(\gamma_0\) is presented for various constant values of \(v\). For convenience it is assumed that at \(\gamma_0=0\) the quantities \((1:n_1)\) and \((1:n_2)\) are equal to identical segments for all values of \(v\). The curves in the figure thus characterize the change in the shape of the wave front with increasing \(v\) (and consequently also the degree of ionization of the layer \(N_0\)), i.e. with approach, as we shall see below, to the place of reflection of each of the waves. It is evident from the figure that the shape of the front of each of the waves differs substantially from a spherical one, and that in the region of reflection the ordinary wave has a saddle-like form, while the extraordinary wave has an elongated elliptical form.

Formula (2,7) plays a very large role in all investigations of the ionosphere; therefore we shall consider its properties in somewhat greater detail. We recall that (2,7) was derived for the case \(\nu=0\), i.e. without taking absorption in the medium into account. If one takes \(\nu\ne0\), then for those values of \(\frac{\nu}{\omega}\) most often encountered in the ionosphere, the properties of \(n^2\) differ little from the properties of expression (2,7), in which \(\nu\) is not taken into account.

From the analysis of formula (2.7) it is seen (cf. Figs. 2, 3, 4) that, with increasing \(v\), i.e., with the degree of ionization of the medium, the values \(n_1^2\) (solid lines in the figures) and \(n_2^2\) (dashed lines) initially decrease from a value equal to unity to a value equal to zero.

The refractive index \(n_1\) has, over the entire frequency range, one value equal to zero at

\[ v_{01}=1. \tag{2.8} \]

The only exception is the case \(\gamma=1\), i.e., longitudinal propagation, when \(n_1^2=0\) at \(v=1+h\).

Fig. 1.

Fig. 2.

The refractive index \(n_2^2\) is equal to zero for \(h<1\) at the values

\[ v_{02}=1-h \quad \text{and} \quad v_{03}=1+h, \tag{2.9} \]

and for \(h>1\) at the value \(v=1+h\).

For \(\gamma=1\), \(n_2^2=0\) only at \(v_{02}=1-h\).

It is also seen from the figures that both \(n_1^2\) and \(n_2^2\) in some cases undergo a discontinuity, which disappears and turns into a sharp maximum when \(\nu \ne 0\). The values of \(v\) at which \(n^2=\infty\) are determined for \(n_1\) and \(n_2\) by one and the same formula

\[ v_{\infty}=\frac{1-h^2}{1-h^2\gamma^2}. \tag{2.10} \]

In this case \(n_1=\infty\) in the frequency range determined from the condition

\[ 1<h\cos\gamma_0, \tag{2.11} \]

and \(n_2=\infty\) when

\[ 1>h. \tag{2,12} \]

The values of \(v\) for which \(n_{1,2}=0\) are important for various calculations, since they determine the magnitude of the degree of ionization \(N_0\) at which the corresponding wave is reflected. This is connected with the fact that, when a wave propagates in a medium inhomogeneous along the \(Z\) axis along \(Z\), as occurs in the ionosphere when a wave is incident vertically on a layer, its complete reflection occurs in the case \(v=0\) when

Fig. 3.          Fig. 4.

\(n=0^{23,24,25}\). Taking absorption into account\(^{26}\) \((\nu \ne 0)\), reflection of the wave occurs when

\[ n^2-\chi^2=0, \]

where \(\chi\) is the absorption coefficient—the imaginary part of the complex refractive index

\[ n^*=n-i\chi. \]

From the system of equations (2,6) one can also calculate the relations between the various components of the electromagnetic field. It is easy to show, for example, that in the plane of the wave front the ratio between the components of the electric induction vector is equal to

\[ \frac{D_x}{D_y} = i\,\frac{2h\gamma(1-v)} {h^2(1-\gamma^2)\mp\sqrt{h^4(1-\gamma^2)^2+4h^2\gamma^2(1-v)^2}} = iM_{1,2}. \tag{2,13} \]

(The \(x\) axis lies in the plane \((NH_0)\) and is perpendicular to \(\mathbf N\).)

Analogous expressions can be obtained for the ratio of the components of the electric and magnetic field vectors.

It is seen from (2,13) that both the ordinary and the extraordinary waves are elliptically polarized, and moreover they have different signs of rotation. In the case where the angle between \(\mathbf N\) and \(\mathbf H_0\) is less than \(\frac{\pi}{2}\), thus for—

the so-called ordinary wave [the upper sign at the radical in (2,13)] has left-hand rotation of polarization, while the extraordinary wave has right-hand rotation. It also follows from (2,13) that

\[ M_1=-\frac{1}{M_2}, \tag{2,14} \]

where the indices (1) and (2) refer respectively to the ordinary and extraordinary waves.

Fig. 5.

Fig. 6.

In Figs. 5 and 6 the dependence of \(M_1\) (solid line) and \(M_2\) (dashed line) on \(v\) [i.e., on the degree of ionization of the layer \(N_0\)—see (2,5)] is shown for two different values of \(h\) in the case \(\gamma>0\). It is seen from the figures that at \(v=1\) (the region of reflection of the ordinary wave) both waves become linearly polarized; moreover, for \(v>1\) the signs of rotation of the polarization of both waves change to the opposite ones.

In experiments it is important to know the state of polarization of the wave reflected from the ionosphere at its exit from the ionosphere. Exact calculations show \(^{27,28}\) that in this case correct results are obtained if in formula (2,13) one lets \(v\to 0\) (i.e., if the limiting transition of the geometrical-optics approximation is applied). Figure 7 presents the results of the corresponding calculation of \(M_2\) [by formula (2,13)] as a function of the angle \(\gamma_0\) between the normal \(\mathbf{N}\) to the wave front and the vector of the earth’s magnetic field \(\mathbf{H}_0\), for various values of \(h\).

From the preceding it is seen that the polarization ellipse of the ordinary wave is oriented so that its major semiaxis is directed along the projection of the vector \(\mathbf{H}_0\) onto the plane of the wave front; in this same direction lies the minor semiaxis of the ellipse of the extraordinary wave.

b) Trajectory of a wave in the ionosphere

For the analysis of the propagation of radio waves in the ionosphere, the geometrical-optics approximation proves applicable almost up to the very point of reflection of the wave \(^{21,23}\). This means that, when consid-

...of the wave trajectory in the layer, one may, with a large degree of accuracy, use the ray interpretation of the problem.

Each of the ionospheric layers is a medium inhomogeneous with height, i.e., in each of them the degree of ionization initially increases with height from values \(N_0\), close to zero at the beginning of the layer, reaches a maximum \(N_m\) at some height, and then decreases. Since the refractive index is less than unity and decreases with height (see Figs. 2–4) from a value equal to unity at the beginning of the layer, the front of the wave incident on the layer is gradually curved and reaches a region where the ray becomes horizontal. At this point

\[ n=\sin \chi_0 \tag{2.15} \]

(where \(\chi_0\) is the angle of incidence of the wave at the beginning of the layer), since along the entire path of the wave Snell’s law is satisfied for the wave front:

\[ n\sin\chi=\sin\chi_0=\text{const}. \]

Fig. 7.

Subsequently the ray returns back to the earth along a trajectory symmetrical (with respect to the point of reflection) to the path by which the wave entered the layer. In the case of vertical incidence, the ray penetrates into the layer as far as the region \(n=0\), where the wave turns back. In both cases, provided that the values \(n=\sin\chi_0\) or \(n=0\) are reached at a distance of several wavelengths or more from the maximum of the layer (which depends on the frequency of the incident wave), the reflection is almost complete (if \(\nu=0\)), i.e., the reflection coefficient is close to unity.

This simple picture describes quite correctly the trajectory of a wave in an inhomogeneous medium of the ionospheric type in the case when the earth’s magnetic field is neglected. When the earth’s magnetic field is taken into account, the picture becomes more complicated. Quite apart from the fact that the wave then splits into two—ordinary and extraordinary—which propagate along different trajectories and are reflected at different places in the layer, each of these trajectories has a more complex form. This is explained by the complex dependence of the refractive index on the angle between the normal to the wave front and the direction of the earth’s magnetic field, the magnitude of which changes continuously along...

along the entire path of propagation of the wave in the layer. In the present case the trajectory of the wave cannot be characterized by the normal \(\mathbf N\) to the wave front, because the direction \(\mathbf N\) does not coincide with the direction of the energy flow—the Poynting vector, averaged over time. The difficulties that arise in this case are already clearly seen from the kinematics of the normals to the front of an infinite plane wave, whose trajectory is shown in Fig. 8\({}^{29}\). In addition, in experiment we always deal with a signal (pulse), with a bounded sinusoid, i.e., with a group of monochromatic waves, and also with a plane wave bounded in space; one may say, with a “piece” of a plane wave or with a spherical wave. Therefore, in order to determine the trajectory of a signal, it is necessary in the present case to consider the propagation of a nonmonochromatic group of waves\({}^{30}\). The problem reduces to finding how, along the path of propagation of the signal, the direction of the vector of its group velocity changes; this direction does not coincide (as in the isotropic case) with the direction of the normal to the wave front, but coincides with the time-averaged Poynting vector (see \({}^{30,31}\)).

Fig. 8.

The group-velocity vector, as is known, is equal to

\[ \mathbf U=\frac{d\omega}{dk} =\frac{d\omega}{dk_x}\mathbf x_0+ \frac{d\omega}{dk_y}\mathbf y_0+ \frac{d\omega}{dk_z}\mathbf z_0, \tag{2,16} \]

where

\[ \mathbf k'(k_x,k_y,k_z)=\frac{\omega}{c}\,n(\omega,\gamma)\mathbf N \tag{2,17} \]

is the wave vector, and \(\mathbf x_0,\mathbf y_0,\mathbf z_0\) are, respectively, the unit vectors of the coordinate system. From (2,16) it follows that the magnitude of the group velocity is

\[ U=\left|\frac{d\omega}{dk}\right|= \frac{ \sqrt{\,1+\dfrac{1-\gamma^2}{n^2}\dfrac{dn}{d\gamma}\,} }{ \dfrac{1}{c}\dfrac{\partial(\omega n)}{\partial\omega} }, \tag{2,18} \]

and the cosine of the angle between the vector \(\mathbf U\) and the normal \(\mathbf N\) to the wave front is equal to

\[ \frac{(\mathbf U\mathbf N)}{U} =\cos(\mathbf N\mathbf U) = \frac{1}{ \sqrt{\,1+\dfrac{1-\gamma^2}{n^2}\left(\dfrac{dn}{d\gamma}\right)^2\,} }. \tag{2,19} \]

In formulas (2,18) and (2,19) the notation introduced above has been used.

It is known that in an isotropic medium the group velocity is equal to

\[ U_0=\frac{1}{\frac{1}{c}-\frac{\partial (kn)}{\partial \omega}} . \tag{2.20} \]

From (2.18) and (2.19) it is seen that

\[ \frac{U_0}{U}=\cos(\mathbf{NU}), \tag{2.21} \]

whence it follows that the group delay time of a signal in the ionosphere, equal in the approximation of geometrical optics to

\[ t_{\mathrm{gr}}=\int \frac{ds}{U} =\int \frac{dl}{U\cos(\mathbf{NU})} =\int \frac{dl}{U_0}, \tag{2.22} \]

is calculated by a formula identical with the formula for the case of an isotropic medium. In (2.22) \(ds\) is the length element in the direction \(\mathbf U\)—the direction of propagation of the signal, and \(dl\) is the length element in the direction of the normal \(\mathbf N\) to the wave front.

At the point of reflection of the wave the tangent to the ray trajectory is horizontal, and this means that at this point

\[ U_z=\frac{\partial \omega}{\partial k_z}=0. \tag{2.23} \]

From (2.23) one obtains the condition for reflection of a wave in the ionosphere in the form

\[ n=\frac{\partial n}{\partial \gamma}\left(\frac{1-\gamma^2}{\gamma}\right). \tag{2.24} \]

Fig. 9.

Expression (2.24), as is seen, differs from the condition for reflection of a wave (2.15) under oblique incidence on a layer in an isotropic medium.

The formulas (2.18) and (2.19) given above made it possible to analyze\({}^{30}\) the trajectory of a signal in the ionosphere, which has a number of features distinguishing it from the trajectory of a ray in an inhomogeneous isotropic medium.

The trajectory of a signal in the ionosphere is a spatial, not a plane, curve. The ray leaves the plane of incidence \((xz)\)—(see Fig. 9). The form of the trajectory depends essentially on the angle between the plane of incidence of the wave and the vector \(\mathbf H_0\) of the magnetic field.

earth. The specific features of the ray trajectory in the ionosphere are evident from Figs. 10–13, which show projections of various ray trajectories onto all coordinate planes, respectively for the ordinary and extraordinary waves. The curves in the figures were constructed

\[ \chi_0 = 5^\circ \text{ (for } m_1\text{)} \]

Fig. 10.

\[ \chi_0 = 5^\circ \text{ (for } m_1\text{)} \]

Fig. 11.

for different values of the angle of incidence \(\chi_0\) and of the angle \(\psi_H\) between the horizontal component of the earth’s magnetic field and the plane of incidence. From the figures it is clear that the ray trajectory is asymmetric with respect to the point of reflection, so that the ray does not return to the plane of incidence. This circumstance leads to the fact that, in the horizontal plane, the direction of arrival of the wave does not coincide with the line connecting the receiver and the transmitter. These properties of the ray

lead to the fact that the wave reflected from the ionosphere does not retain the spherical form of the incident wave, but has an elliptical form in the horizontal plane.

For vertical incidence of the wave, the ray propagates in the plane \((H_0N)\); conversely, the wave propagates along the same trajectory and emerges at the same point of the layer at which it entered the layer.

The above-indicated features of the ray trajectory are manifested strongly at relatively short distances from the transmitter, for angles of incidence on the layer of the order of \(2—10^\circ\). With increasing angle of incidence, the ray trajectory approaches more and more the symmetric trajectory of a wave in an isotropic inhomogeneous medium and only slightly leaves the plane of incidence.

c) Spreading of the pulse

In conclusion of the present paragraph let us consider how the form of a signal (pulse) changes in the ionosphere.

In order to describe the propagation of a signal, it is convenient to represent it, as is known, in the form of an integral or Fourier series, giving the decomposition of the pulse into sinusoids, i.e. into monochromatic waves. If the refractive index \(n=n(\omega)\), i.e. the medium is dispersive, as is the case in the ionosphere [see (2.7)], then a pulse propagating in such a medium is deformed. This is explained by the fact that the values of the phase velocity

\[ v=\frac{c}{n} \]

of its individual components are different.

For the analysis of signal propagation in a dispersive medium, the concept of a quasimonochromatic group of waves is introduced, meaning by this a monochromatic wave with amplitude and phase varying slowly (in comparison with the carrier frequency of the wave). In this case one considers the propagation of the quasimonochromatic group (the envelope of the high-frequency oscillations), introducing the concept of the group velocity of such a signal

\[ U=\frac{d\omega}{dk}. \]

Fig. 12.

The group velocity is then determined from the fact that, in the expansion in a Taylor series of the wave number \(k=\dfrac{\omega}{c}n(\omega)\),

in the neighborhood of the carrier frequency \(\omega_0\), only the first-order term is used, i.e., one takes

\[ k = k_0 + \left(\frac{dk}{d\omega}\right)_{\omega=\omega_0}\Delta\omega, \tag{2,25} \]

where \(\Delta\omega=\omega-\omega_0 \ll \omega_0\) is a small quantity determining the spectral width of the signal. In this approximation the signal envelope is displaced with the group velocity \(U\), without being deformed.

The restrictions imposed in introducing the concepts of a quasi-monochromatic group and group velocity are well satisfied for those cases which are usually encountered in ionospheric studies. Therefore, when calculating the delay time of a signal and the trajectory of a ray in the ionosphere, it is always assumed that the signal propagates in it with the group velocity \(U\).

However, as has already been indicated above, in this approximation the signal is not deformed. At the same time, for a complete analysis of the propagation of a pulse in the ionosphere it is of interest (and also has, in some cases, practical significance) to consider how it spreads out.

Fig. 13.

To analyze this question it is necessary to take into account terms of higher order in the expansion (2,25); in this case it proves sufficient to take

\[ k = k_0 + \left(\frac{dk}{d\omega}\right)_{\omega_0}\Delta\omega + \frac{1}{2}\left(\frac{d^2 k}{d\omega^2}\right)_{\omega_0}(\Delta\omega)^2 . \tag{2,26} \]

For a pulse specified at the beginning of the medium in the form of a truncated sinusoid and having width \(T\), one obtains, after analysis of the correspond-

tingly computed Fourier integral (see 29), that its amplitude \(A\) (equal to unity at the beginning of the layer) varies with time in the following way:

\[ A(t)=\frac{1}{\sqrt{2}}\left|F\left(\frac{t'}{\sqrt{\pi\,\dfrac{d^2\varphi}{d\omega^2}}}\right) -F\left(\frac{t'}{\sqrt{\pi\,\dfrac{d^2\varphi}{d\omega^2}}} -\frac{T}{\sqrt{\pi\,\dfrac{d^2\varphi}{d\omega^2}}}\right)\right|. \tag{2,27} \]

In formula (2.27), \(F(\ldots)\) is the Fresnel integral, whose upper limit is equal to the quantity in parentheses, and whose lower limit is zero. The quantity

\[ t'=t-\frac{d\varphi}{d\omega} \tag{2,28} \]

is the time, reckoned from the instant when the front of the incident

Fig. 14.
(Curves labeled \(\omega/\omega_c=0.1\), \(\omega/\omega_c=0.4\), \(\omega/\omega_c=0.7\), and \(\omega/\omega_c=1\); horizontal axis \(\xi/T\).)

pulse, propagating with group velocity \(\dfrac{d\omega}{dk}\) \(\left(\dfrac{d\varphi}{d\omega}\right.\) is the time of group delay of the signal), arrives without spreading at the point of observation, and \(\varphi(\omega)\) is the phase of the wave.

If one uses the formula \(\varphi(\omega)\) for a parabolic layer, calculated in the geometrical-optics approximation and having the form

\[ \varphi(\omega)=\frac{Z_{c1}}{c}\,\omega_c \left\{ 1-\frac{\omega_c^2-\omega^2}{2\omega_c\omega} \ln\frac{\omega_c+\omega}{\omega_c-\omega} \right\}, \tag{2,29} \]

then one can calculate from formula (2.27) how the shape of the pulse in the layer changes as a function of the ratio \(\dfrac{\omega}{\omega_c}\)—the frequency of the wave \(\omega\)

to the so-called critical frequency \(\omega_c\), characterizing the depth of penetration of the wave into the layer (see Fig. 14). It is known that at \(\dfrac{\omega}{\omega_c}=1\) reflection takes place at the maximum of the layer, since \(v=1\) gives \(\dfrac{4\pi N_m e^2}{m}=\omega^2\), and \(\omega_c=\dfrac{4\pi N_m e^2}{m}\)—see (2.8). In formula (2.29) \(Z_{\text{sl}}\) is the half-thickness of the layer.

From Fig. 14 it is seen that at \(\dfrac{\omega}{\omega_c}=0.1\) the pulse is spread very little. With increasing \(\dfrac{\omega}{\omega_c}\) the width of the pulse becomes ever greater, and at \(\dfrac{\omega}{\omega_c}=0.98\) the width of the pulse, if it is defined, say, between amplitude values equal to \(0.2\) of the amplitude of the incident pulse (in the figure the incident pulse is shown by a dotted line), becomes approximately equal to \(3.33T\). The calculation was carried out for \(T \approx 3\cdot 10^{-5}\) sec.

One should not attach great importance to the shape of the top of the pulse shown in Fig. 14, which has a complicated form because of the idealization of the calculation.

3. STRUCTURE OF THE IONOSPHERE

Soon after the existence of the lower layer \(E\) in the ionosphere, lying at a height of about \(100\) km above the earth’s surface, had been experimentally proved by means of the pulse method, a second layer \(F\) was also discovered at a height of \(200\)—\(250\) km\({}^{33}\). The existence of these two principal layers in the ionosphere is clearly seen from a typical height-frequency characteristic (usually obtained in vertical sounding of the ionosphere), shown on the oscillogram (see inset). It is clearly seen how, when the acting frequency is varied, the effective height \(Z_0\) (or, equivalently, the group delay time of the signal) initially remains almost unchanged upon reflection from the layer \(E\) (white line \(a\) in the figure), then a certain increase of \(Z_0\) is observed and a jump-like change of \(Z_0\) in the region of the critical frequencies \(f_E^\circ\) of the layer \(E\). This corresponds to the transition of reflection from the maximum of layer \(E\) to the beginning of layer \(F\). The effective height, upon reflection from layer \(F\), also at first changes little (branch \(b\) in the figure); however, with increasing frequency and with the penetration of the wave into deeper regions of the layer, the rate of growth of \(Z_0\) increases, and on approaching the maximum of the layer a splitting of the signal into the ordinary and extraordinary is observed. The cessation of reflection of the ordinary wave occurs at values of the critical frequency

\[ f_F^{02}=\frac{4\pi c^2 N_m e^2}{(4\pi)^2 m}, \tag{3.1} \]

where \(N_m\) is the degree of ionization of the maximum of the layer [see (2.8)]. Thus the layer becomes transparent for the ordinary wave at

I

II

III

Top diagram: ionogram with vertical axis labeled \(Z_0\), km, marked

\(0,\ 200,\ 400,\ 600,\ 800,\ 1000,\ 1200\),

and horizontal axis labeled \(f\), MHz, marked approximately

\(0.5,\ 0.6,\ 1,\ 1.4,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12,\ 13,\ 14,\ 15,\ 16\).

Visible curve labels: \(E\), \(F_1\), \(F_2\).

IV

Lower set of ionogram strips.

Visible repeated vertical label at left: \(Z_0\).

Visible handwritten annotations above the strips include: \(\infty h/2\), \(-\infty h/2\), and \(0 h/2\) [[unclear: handwritten notation]].

Bottom horizontal label: “Frequency in MHz.”

frequencies \(f \gg f_F^0\). For the extraordinary wave this occurs [see (2.9)] at the frequency

\[ f_F^x = f_F^0 + \frac{f_H}{2} \tag{3,2} \]

\[ \left(\text{for } f \gg \frac{f_H}{2}\right), \]
where \(f_H=\dfrac{eH_0}{2\pi mc}\) is the gyrofrequency.

Thus we see that double refraction in the \(F\) layer leads to two-tailed height–frequency characteristics of the ionosphere.

The observed double refraction in the \(F\) layer indicates that the principal charge carriers of this layer, which influence the propagation of radio waves, are electrons, since in the case of a substantial influence of ions the gyrofrequency \(f_H\) would be very small (because of the large mass of ions) and the splitting of the signal would be difficult to detect.

For a long time it was therefore assumed that in the \(E\) layer the most active agent influencing the propagation of radio waves was the ions. At present, however, much experimental evidence has already been obtained indicating that not only in the normal \(E\) layer is there double refraction\(^{34}\), but also that ordinary and extraordinary waves are observed in the \(E_{\text{spor}}\) layer\(^{35,36}\). One reason why in \(E\) a splitting of the signal into two is not always observed is the small thickness of the layer and the large gradients of its ionization. This leads to the fact that the difference in the group times of the ordinary and extraordinary signals is small and both signals merge on the oscillogram. In addition, reflections from the very frequently observed \(E_{\text{spor}}\) smear the edge of the characteristic of the \(E\) layer.

Branch 2 on oscillogram I corresponds to double reflection from \(F\).

It was subsequently found that in summer the \(F\) layer splits into two layers, \(F_1\) and \(F_2\). The height–frequency characteristics then have the form shown in oscillogram II.

In addition, it was established that reflections from the \(E\) and \(F\) layers are observed quite often simultaneously at frequencies exceeding the critical frequency of the \(E\) layer, and sometimes also at frequencies greater than the critical frequency of \(F\). In this case the ionospheric characteristic has the form shown in oscillogram III. These reflections indicate that regions of increased ionization appear in the \(E\) layer—as though clouds embedded in it—so that the layer remains transparent to signals whose frequency exceeds its critical frequency. At the same time these clouds cause a fairly stable partial reflection of the wave incident on the layer. Often this accumulation of clouds, called \(E_{\text{spor}}\), completely screens the \(F\) layer.

The simultaneous appearance of reflections from \(E\) and \(F\) may also be a consequence of an increase in the ionization gradient of layer \(E\). However, the often observed complete screening of \(F\), as well as the double-ray refraction of \(E_{\mathrm{spor}}\), speak rather in favor of the idea that \(E_{\mathrm{spor}}\) is a region of increased ionization.

Recently, from a number of experimental data\(^{37}\) it follows that in layer \(F\) there also, apparently, appears in some cases an \(F_{\mathrm{spor}}\) of ordinary structure, which differs from \(E_{\mathrm{spor}}\) in that it, like layer \(F\), has a large thickness. In addition, the ionization in \(F_{\mathrm{spor}}\) exceeds the ionization in \(F\) by only 30–40%. The ionosphere characteristic in this case has the form shown on oscillograms IV, i.e., instead of 2 tails there appear 4 (and sometimes even 6 tails).

In some cases, three-tailed characteristics are observed in layer \(F_2\),\(^{38,39,40}\) in which the third tail may, under certain conditions, appear not from \(F_{\mathrm{spor}}\), but owing to the leakage of part of the wave energy above the region of reflection of the ordinary wave (where \(n_1 = 0\)). In this case reflection occurs at \(v = 1 + n\), where \(n_2\) is secondarily equal to zero. (For an analysis of this phenomenon see works \(^{39,41,42}\).)

Below layer \(E\) there is also a region of increased ionization; moreover, owing to the large number of collisions, it has a substantial effect on the intensity of radio waves reflected from the ionosphere. This region of the ionosphere has no sharply expressed maximum, and therefore regular reflections from it are not observed; it is customary to call it layer \(D\).

Along with the layers indicated above, more or less prolonged reflections from various heights of the ionosphere are often observed. These reflections are, in the main, of a random character and appear most often during ionospheric storms from ionized clouds arising in the ionosphere.

4. DIURNAL AND SEASONAL VARIATION OF THE IONOSPHERE

DATA ON THE VARIOUS LAYERS

Up to the present time, the principal quantity by means of which the ionosphere is characterized and by the variation of which one establishes one or another regularity of the various layers, and also traces their connection with other geophysical phenomena or phenomena on the Sun, is the critical frequency of each of the layers. The critical frequency, as we have seen [see (2,8) and (3,1)], determines the value of the maximum ionization of the layer, equal to

\[ N_{\mathrm{m}} = 1.24 \cdot 10^4 f_c^2\ \text{cm}^{-3}. \]

Another quantity by which the various layers of the ionosphere are usually characterized is the minimum value of the effective height, corresponding to the height of the beginning of each of the layers.

In Figs. 15 and 16 typical averaged curves of the diurnal variation of ionospheric characteristics are given for the middle latitudes of the northern hemisphere, for winter and summer conditions. The values of the critical frequencies \(f_c\) and of the minimum effective heights \(Z_m\) have only slight fluctuations relative to their mean value for the \(E\) layer, whereas for the \(F\) layer the deviations from the mean value reach 10–15% and more even on quiet, undisturbed days.

Fig. 15.

The large accumulated experimental material on diurnal and seasonal changes of the characteristics has led to a number of conclusions concerning appreciably different layers of the ionosphere.

a) Data for the \(E\) layer

The minimum effective heights of the \(E\) layer have no definite dependence on the time of day or year and vary within the limits 110–120 km. The ionization distribution of the \(E\) layer is often well approximated by a parabola with a half-thickness equal to 15–20 km.

The critical frequency \(f_E\) has a diurnal variation, well repeated at different times of the year, with a maximum at local noon; the maximum value of the ionization \(E\) is greater in summer than in winter.

The change in the electron concentration of the \(E\) layer is described rather well by the equation

\[ \frac{dN}{dt}=J_0-\alpha N^2, \tag{4,1} \]

where \(J_0\) is the number of electrons formed in \(1\ \mathrm{cm}^3\) of the layer in one second under the action of an external source of ionization—the radiation of the Sun; and \(\alpha\) is the effective recombination coefficient of the layer. The application of equation (4.1) to the analysis of the variation of \(N\) during solar eclipses \(^{43,44}\), as well as of the diurnal and seasonal variation of the critical frequencies, has led to the conclusion that the principal ionizing agent of the \(E\) layer is the ultraviolet radiation of the Sun. If one proceeds from these data, then one obtains a diurnal dependence (and also a latitudinal one—see Table II below), well confirmed by experiment,

\[ f_E \sim \sqrt[4]{\cos \chi}, \tag{4.2} \]

where \(\chi\) is the zenith distance of the Sun. On the assumption that there is no seasonal dependence of \(\alpha\) and of the temperature of the layer, it also follows that the ratio of the summer noon values of the maximum ionization to the winter values must satisfy the relation

\[ \frac{N_l}{N_z} \sim \sqrt{\frac{\cos \chi_l}{\cos \chi_z}}, \tag{4.3} \]

which is likewise well confirmed by experimental data (see Table I) for various points of the globe.

Fig. 16.

It follows from various measurements that the recombination coefficient \(\alpha\) varies for the \(E\) layer within the limits \((0.5 \div 1.2)\cdot 10^{-8}\) by day and \((1 \div 4)\,10^{-9}\) by night.

The scant data indicate that the mean value of the collision frequency \(\nu\) for the \(E\) layer is of the order of \(10^6\ \mathrm{sec}^{-1}\).

From an analysis of data for the \(E\) layer over a long period it has been established that the variation of the monthly mean values of the number of sunspots and of the noon values of the critical frequencies of the \(E\) layer agree well, even in

in a number of details following the eleven-year cycle of solar activity.

Table I

Place of observations	Latitude	$\dfrac{N_4}{N_3}$ — experimental	$\dfrac{N_4}{N_3}$ — theoretical
Tomsk	$56^\circ 30$	2.21 (1941—1945)	2.19
Slough (England)	$51^\circ 30$	1.90 (1934—1938)	1.85
Deal (USA)	$40^\circ 15$	1.50 (1933—1934)	1.47
Washington	$38^\circ 55$	1.44 (1933—1937)	1.44
Hiraiso (Japan)	$36^\circ$	1.37 (1936)	1.36
Watheroo (Australia)	$30^\circ 19$	1.25 (1939—1941)	1.29
Huancayo (Peru)	$12^\circ 02$	1.17 (1938—1941)	1.10

Table II gives the monthly mean obtained values of the maximum values of the degree of ionization of layer $E$ for the summer and winter months for different latitudes, respectively for the time of the minimum and maximum of the eleven-year period of solar activity. From the table one can see within what limits $N_m$ varies at different points of the globe.

Table II

Latitude $\varphi$	Values of $N_m$ (electrons in $1\ \mathrm{cm}^3$) — Minimum of solar activity, July	Values of $N_m$ (electrons in $1\ \mathrm{cm}^3$) — Minimum of solar activity, February	Values of $N_m$ (electrons in $1\ \mathrm{cm}^3$) — Maximum of solar activity, July	Values of $N_m$ (electrons in $1\ \mathrm{cm}^3$) — Maximum of solar activity, February
$69^\circ 40$	$1.1\cdot 10^5$	—	$1.7\cdot 10^5$	$0.75\cdot 10^5$
$56^\circ 30$	$1.34\cdot 10^5$	$0.78\cdot 10^5$	$2.0\cdot 10^5$	$1.10\cdot 10^5$
$38^\circ 50$	$1.44\cdot 10^5$	$1.1\cdot 10^5$	$2.16\cdot 10^5$	$1.74\cdot 10^5$
$-12^\circ 02$	—	—	$2.0\cdot 10^5$	$2.42\cdot 10^5$
$-30^\circ 19$	—	—	$1.61\cdot 10^5$	$2.0\cdot 10^5$

b) Data for layer $F_1$

It has already been indicated above that in the daytime in summer layer $F$ splits into two: $F_1$ and $F_2$. As a result, the characteristics of layer $F_1$ can be considered only over a limited interval of time.

The minimum height of $F_1$ usually decreases toward noon, reaching values of 220—230 km.

The ionization of the \(F_1\) layer follows the same course as the ionization of \(E\), and is well described by formulas (4.1) and (4.2), so that it exhibits the same diurnal, seasonal, latitudinal, and eleven-year dependences. The recombination coefficient of \(F_1\), according to the data of a few measurements, is equal to \((2 \div 8)\cdot 10^{-9}\).

Table III gives the monthly mean midday values obtained for the maximum ionization \(N_{\mathrm{m}}\) of the \(F_1\) layer for a summer month, respectively for the times of minimum and maximum of the eleven-year period of solar activity.

Table III

Latitude \(\varphi\)	Values of \(N_{\mathrm{m}}\) (electrons in \(1\ \mathrm{cm}^3\)) Minimum of solar activity July	Values of \(N_{\mathrm{m}}\) (electrons in \(1\ \mathrm{cm}^3\)) Maximum of solar activity July
\(63^\circ 40\)	\(2.1\cdot 10^5\)	\(3.1\cdot 10^5\)
\(56^\circ 30\)	\(2.3\cdot 10^5\)	\(3.85\cdot 10^5\)
\(38^\circ 60\)	\(1.55\cdot 10^5\)	\(3.5\cdot 10^5\)
\(-12^\circ 02\)	—	\(3.6\cdot 10^6\)
\(-30^\circ 19\)	—	\(3.15\cdot 10^5\)

c) Data for the \(F_2\) layer

The behavior of the \(F_2\) layer is considerably more complicated than that of the \(E\) and \(F_1\) layers. The minimum values of the effective heights \(F_2\) after sunset have approximately the same values both in summer and in winter, and vary within the limits of 250–300 km. In the daytime in summer the height of the \(F_2\) layer reaches (approximately at noon) a maximum of the order of 350–400 km and more. In winter, on the contrary, the height of \(F_2\) has at noon a minimum of the order of 230–250 km.

The thickness of the \(F_2\) layer and the distribution of its ionization with height are subject to considerably greater variations. At present there is not a sufficient number of systematically processed data for these quantities. However, individual values obtained in processing the characteristics of the \(F_2\) layer show that over the course of a day the semi-thickness of the layer changes, in some cases, from 80–100 km to 200–250 km.

The diurnal characteristics of the \(F_2\) layer also differ significantly in summer and in winter. In winter the critical frequency has a single maximum close to noon. At the same time, however, the course of ionization does not follow

according to the law \(f_{F_2}\sim \sqrt[4]{\cos\chi}\). In summer, on the contrary, the critical frequency \(F_2\) often has a minimum at noon and two maxima asymmetrically situated relative to it, one of which—the pre-evening one—is more pronounced and, often, is the only maximum during the day.

The seasonal course of the maximum noon values has, in contrast to \(f_E\) and \(f_{F_1}\), a minimum in summer (in June–July), and two maxima: one in February–March and the second in October–November.

Fig. 17

During solar eclipses the course of ionization \(F_2\) also does not strictly follow the course of the Sun’s ultraviolet radiation. The ionization minimum, as a rule, is insufficiently deep and does not correspond to that expected if, in the calculations, one proceeds from the usual conceptions—using equation (4.1). In addition, this minimum is displaced relative to the center of the eclipse—it occurs in the second half of the eclipse.

Determination of the recombination coefficient \(\alpha\) is possible for the \(F_2\) layer from the processing of measurement data during sunrise and sunset or at night, and in winter during hours close to noon. The results of individual calculations give the following values of \(\alpha\) for the \(F_2\) layer: by day \(\alpha\) varies within \((4\div 10)\cdot 10^{-11}\), and at night within \((0.6\div 8)\cdot 10^{-10}\).

At the same time, a general analysis of the variations in ionization of the \(F_2\) layer indicates that the ultraviolet radiation of the Sun is still the principal ionizing agent. Moreover, in recent works\(^{15}\)

it has been shown that the ionization of the \(F_2\) layer, like that of the \(E\) and \(F_1\) layers, as well as the amplitude of the Earth’s magnetic field, are apparently subject to the influence of the same regions of the Sun’s ultraviolet radiation. The mean values of the critical frequencies \(F_2\) have a linear variation depending on the number of sunspots and, if not in every detail, then in any case in general they follow quite well the eleven-year cycle of solar activity.

The situation is also complicated with the latitudinal dependence of the \(F_2\) layer. The composition of the \(F_2\) layer apparently changes from place to place, and one may speak of the climate of the \(F_2\) layer, different at different points of the terrestrial globe\(^{16}\). In some

Fig. 18.

works\(^{45}\) a connection is indicated between the critical frequencies and meteorological conditions—monthly mean values of pressure at the Earth’s surface; in other investigations a rather curious connection is noted between \(f_{F_2}\) and the inclination and latitude of the Earth’s magnetic field\(^{13,17}\), etc. At present, searches are being conducted in various directions for general laws explaining the phenomena occurring in the \(F_2\) layer.

The longitudinal difference in the ionization of the \(F_2\) layer is clearly visible from the ionization maps shown in Figs. 17, 18, on which lines of equal magnitude of the daily values of ionization for April 1945 are plotted, respectively for \(300\)—\(60^\circ\) and \(120\)—\(240^\circ\) geomagnetic longitude.

Table IV gives the monthly mean midday values of the ionization \(N_{\mathrm{m}}\) of the \(F_2\) layer for summer and winter months and various latitudes for the minimum and maximum of the eleven-year period of solar activity.

According to individual measurements, the mean number of collisions in the \(F_2\) layer varies within the limits \((1 \div 50)\cdot 10^3\).

Table IV

Latitude \(\varphi\)	Longitude \(\lambda\)	\(N_m\) value (electrons per \(\mathrm{cm}^3\)): solar-activity minimum, July	\(N_m\) value (electrons per \(\mathrm{cm}^3\)): solar-activity minimum, February	\(N_m\) value (electrons per \(\mathrm{cm}^3\)): solar-activity maximum, July	\(N_m\) value (electrons per \(\mathrm{cm}^3\)): solar-activity maximum, February
\(69^\circ 40\)	\(18^\circ 55\) W.	\(2.7\cdot 10^5\)	\(3.2\cdot 10^5\)	\(5.9\cdot 10^5\)	\(9.8\cdot 10^5\)
\(56^\circ 30\)	\(84^\circ 54\) E.	\(3\cdot 10^5\)	\(5.2\cdot 10^5\)	\(8.1\cdot 10^5\)	\(15.5\cdot 10^5\)
\(38^\circ 50\)	\(77^\circ 00\) W.	\(3.3\cdot 10^5\)	\(6.9\cdot 10^5\)	\(6.4\cdot 10^5\)	\(17.5\cdot 10^5\)
\(-12^\circ 02\)	\(75^\circ 20\) W.	—	—	\(9.3\cdot 10^5\)	\(18.5\cdot 10^5\)
\(-30^\circ 19\)	\(115^\circ 52\) E.	—	—	\(11.0\cdot 10^5\)	\(12.4\cdot 10^5\)

г) Data on the sporadic \(E\) layer

It has already been pointed out above that, under certain conditions in the \(E\) layer, there apparently arise accumulations of ionized clouds of enhanced ionization, fairly extended spatially and stable in time (several hours or more), forming the so-called sporadic \(E\) layer. The \(E_{\mathrm{spor}}\) layer appears rather often, and reflections from it \((f \gg f_E)\) are so intense and stable that radio communication often proceeds mainly by means of them.

The physical nature of \(E_{\mathrm{spor}}\) remains unclear up to the present time. There are no definite data that would make it possible to draw conclusions about the causes of its formation, about the source of its ionization, etc. At the same time, some of its properties indicate that the ionizing agent is apparently, after all, of solar origin.

The appearance of \(E_{\mathrm{spor}}\) is local and temporal in character, so that it is often not observed simultaneously at different points separated from one another by several tens of kilometers.

During the course of a day, \(E_{\mathrm{spor}}\) appears most often closer to midnight, although it is more intense (in degree of ionization) during the day. According to some data, the degree of ionization of \(E_{\mathrm{spor}}\) reaches, in some cases, \(5\cdot 10^6\ \mathrm{el.}\ \mathrm{cm}^{-3}\). Investigation of \(E_{\mathrm{spor}}\), both with the aid of a polarimeter and by vertical sounding\({}^{36}\), has shown that double refraction is observed in it. This permits the assumption that its ionization is of electronic origin.

The effective height of \(E_{\mathrm{spor}}\) is almost unchanged at all frequencies and co-

lies within 100–110 km, which is explained by its small thickness, which, according to the latest data, varies within 50–500 m.

The seasonal course of \(E_{\mathrm{spor}}\) coincides with the seasonal course of the normal layer \(E\), i.e. it appears more often in the summer months \(^{46,47,48}\). As for the eleven-year cycle, the data available in the literature are contradictory. According to measurements in the northern hemisphere, it has been established that the intensity and frequency of occurrence of \(E_{\mathrm{spor}}\) seem to decrease with increasing solar activity. On the other hand, for the southern hemisphere \(^{46}\) an indefinite correspondence has been obtained between the cycle of solar activity and the course of \(E_{\mathrm{spor}}\). In the first half of the decline of solar activity—from 1938 to 1941—an increase of \(E_{\mathrm{spor}}\) was observed, from 1941 to 1944—a decrease, and then—in 1945—an increase of \(E_{\mathrm{spor}}\) began again.

No correspondence is observed between \(E_{\mathrm{spor}}\) and the 27-day period of the Sun, and there are also no definite indications of an intensification of \(E_{\mathrm{spor}}\) during periods of magnetic activity. At the same time, observational data in northern latitudes indicate an intensification of \(E_{\mathrm{spor}}\) during periods of ionospheric storms and when auroras appear. In addition, it has been established that \(E_{\mathrm{spor}}\) appears more often at middle and high latitudes and more rarely in the equatorial zone.

Individual experiments indicate an intensification of \(E_{\mathrm{spor}}\) during periods of thunderstorm activity. There are also indications of the appearance of \(E_{\mathrm{spor}}\) during the passage of meteorites. All these factors may possibly influence \(E_{\mathrm{spor}}\), concerning the mechanism of whose origin there are as yet no definite considerations.

5. IRREGULAR PHENOMENA IN THE IONOSPHERE

Various types of irregular, sporadic phenomena are observed rather often in the ionosphere. Some of these anomalies, disturbing the state of the ionosphere, change its ionization very substantially and have a long duration—lasting several days and even up to a month. Others are short-lived and have little effect on the general state of the ionosphere. Here we shall briefly characterize the best-known irregularities of the ionosphere.

a) Ionospheric storms

During periods of strong disturbances of the Earth’s magnetic field—during magnetic storms—significant changes in the state of the ionosphere are also observed, called ionospheric storms.

During these storms the state of the ionosphere becomes very unstable. Radio communication is strongly disturbed, especially at high frequencies. In polar regions, in some cases, the interruption of radio communication covers the range from 20 to 1 MHz and beyond. An ionospheric storm usually develops over a period from several minutes to an hour or more. Its duration ranges from several days to

month. Normal ionospheric conditions after a storm are often restored over the course of several days.

Ionospheric storms are strongest in polar latitudes, but they are also observed at the equator. Storms occur, with a slight shift in time, almost simultaneously over a large portion of the Earth. In the polar region, storms are accompanied by intense auroras, which during especially disturbed periods have been observed even at middle latitudes.

Fig. 19.

The connection between ionospheric storms, magnetic storms, and auroras indicates that ionospheric storms are caused by electrically charged corpuscular streams ejected by the Sun, which are turned by the Earth toward its magnetic poles and produce auroras. Ionospheric storms are often observed during periods of increased solar activity.

In the first phase of the development of ionospheric storms, a substantial decrease is noted in the maximum value of the ionization of the \(F_2\) layer*). This is one of the principal signs of an impending storm\(^{49}\). Often a certain decrease in the critical frequency \(F_1\) is also observed. It is characteristic that during an ionospheric storm, when penetrating through the \(F_1\) layer, there is a more considerable effect of an increase in the time

*) The critical frequency \(F_2\) is often generally difficult to determine during a storm.

of group delay than usual. This is evident from Fig. 19, which, for comparison, shows the height–frequency characteristics of the ordinary ray for a disturbed and a magnetically quiet day.

The layered structure of the ionosphere is disrupted during the first few hours of a strong ionospheric storm. The structure of the layers becomes more complicated. Often the entire space between \(E\) and \(F\) is filled with ionized clouds, changing over the course of several minutes or less. The ionosphere becomes turbulent. This leads to a strong blurring

Fig. 20.

and to the appearance of additional reflected signals and to a complex type of height–frequency characteristics (oscillogram, Fig. 20).

In the next phase of development of the ionospheric storm, a strongly absorbing region appears below the \(E\) layer, which often leads to the complete disappearance of the reflected signals. A weakening of the intensity of the reflected signals over a broad frequency range is also one of the characteristic features of an ionospheric storm.

In the third phase of an ionospheric storm, a strong increase in the minimum height of the \(F_2\) layer is observed, reaching, in some cases, heights greater than 500 km. During this period the strong absorption in the ionosphere decreases somewhat. The increase in the virtual heights and the decrease in the critical frequencies \(F_2\) are evident from Fig. 21, which shows the diurnal variation of the critical frequencies \(F_2\) and of its virtual heights during a magnetic disturbance (according to observations in the equatorial zone). The lower part of the figure shows the course of the horizontal \((H)\) and vertical \((Z)\) components of the magnetic field and the declination \((D)\) during the storm.

The increase in the virtual heights of the \(F_2\) layer is also one of the main characteristics of an ionospheric storm.

It is of interest that, in summer, in the daytime hours the disturbance of the ionospheric storm spreads throughout the entire \(F_2\) layer, since

at this time it lies high above the Earth’s surface. In winter, by day, when the height of the layer is considerably smaller, the disturbance is observed chiefly in the region of the maximum of the \(F_2\) layer and does not pass into its lower part.

We shall not dwell here in greater detail on the description of ionospheric storms, whose behavior is especially complex in the polar region.

At present it still does not seem possible to explain the nature of these phenomena. It is possible that the decrease in the critical frequencies

Fig. 21.

of the \(F_2\) layer is connected with its considerable expansion and an increase in its thickness; moreover, the ionizing agent is strongly absorbed in the higher region of the ionosphere—there is a displacement of the layer maximum upward. The decrease of ionization in the lower part of the layer subsequently leads to an increase in the minimum effective heights of \(F_2\).

The intensification of the absorption of radio waves over a broad frequency range, on the other hand, testifies to significant ionization specifically of region \(D\). This therefore indicates a great penetrating capacity of the ionizing agent.

The totality of effects observed during ionospheric storms indicates that the ionizing agent causing them may contain both electrically charged and neutral particles of all possible velocities.

b) Sudden disturbances

One of the most remarkable phenomena in the ionosphere consists in a sudden disturbance of its state, which often causes a complete cessation of radio communication carried out by means of the reflected wave.

The phenomenon proceeds in the following way. In some cases, on the illuminated half of the terrestrial globe, radio communication suddenly ceases completely and simultaneously on most paths of the short-wave range over a wide frequency band.

The duration of this phenomenon ranges from several minutes to an hour or more. Normal conditions are restored more slowly, beginning with the shorter-wave part of the disturbed range. It manifests itself more sharply, both with respect to the depth and abruptness of the decrease in received signal strength and with respect to its duration, in the low-latitude zone—closer to the equator—and, in terms of time of day, closer to noon.

Processing the results of observations of these effects[^60] has shown that, in most cases, simultaneously with the disruption of radio communication, disturbances of the Earth’s magnetic field are observed—chiefly of its horizontal component—and of the gradient of the atmospheric electric field, and that it is accompanied by a bright ultraviolet eruption of the Sun’s chromosphere. From numerous observations it also follows that there are rare cases when the disturbances are not accompanied by eruptions of the Sun. The reverse also occurs.

No definite regularity has been established in the occurrence of these disturbances. There are some indications that they are more numerous in years of maximum solar activity; at the same time, their dependence on the Sun’s spot formation has not been revealed. The annual course of the disturbances has a maximum in summer. Often several disturbances appear one after another. During solar eruptions, an ejection by the Sun of luminous matter is noticeable. On the assumption that this corpuscular stream should cause auroras, it was investigated whether they coincide with radio-communication disturbances. It was established that about 50% of auroras are accompanied, during

preceding them by 24 hours, by a disruption of radio communication. However, on the other hand, only about 10% of radio disruptions coincide with disturbances of the aurorae.

Studies of the ionosphere during sudden disturbances have led to a number of conclusions about the state of the various layers during these periods.

In oscillogram V (see inset) a height diagram is given, taken at a fixed frequency during a disturbance. From the figure

Fig. 22

it is evident that the disturbance was accompanied only by the disappearance of reflections from both layers (breaks in the characteristics \(E\) and \(F\)) and did not cause any noticeable change in their effective heights.

The results of processing observations during a strong disturbance, which lasted about three hours, are shown in Fig. 22. In the upper part of the figure the diurnal variation of the minimum heights of various regions of the ionosphere and their mean values over six days (dashed line) are plotted (with circles). In the lower part of the figure the variation of the critical frequencies and of the minimum values of the frequencies at which reflections were still observed (marked by crosses) is given. Lower frequencies were absorbed in the lower part of the ionosphere, and thus curve \(a\), in a certain sense, characterizes the degree of ionization of layer \(D\), in which, as is known, absorption of radio waves is proportional to \(\frac{1}{\omega^{2}}\). During the disturbance, as is evident from the figure, reflections completely disappeared

tion from \(E\), \(F_1\), and \(F_2\); at the same time the minimum frequencies—the curve \(a\)—increased sharply and even exceeded the values of the critical frequencies \(F_2\). This indicates a very large increase in the ionization of region \(D\), which led to the absorption of radio waves over the entire frequency range, as a result of which reflections from all layers ceased. Usually no substantial change is observed in the ionization and heights of the layers \(F_1\) and \(F_2\) caused by this effect. Some consequence of it is observed in layer \(F\)—an increase in ionization and in the minimum effective height.

The restoration of the normal intensity of the reflected signals depends on the frequency and begins earlier at higher frequencies. This is quite natural, since the enhanced absorption should last longer for the lower frequencies, namely until the ionization of layer \(D\) assumes its usual value.

The occurrence, during eruptions on the Sun, of considerable ionization specifically in region \(D\) is apparently connected with selective absorption of the ionizing agent by the lower part of the ionosphere, the gaseous composition of which is still not precisely known.

c) Cloud structure of the ionosphere

Along with reflections from the sporadic layer \(E\), which, as already noted, are of a more or less regular character, reflected signals are observed from layer \(E\) of considerably lower intensity, of short duration, and at times strongly diffuse. In these reflections there is no regularity in the diurnal or seasonal course, etc.

A number of experimental investigations have made it possible to reveal the features of this type of reflection, as well as their forms in other layers. However, the complexity of the phenomenon, indicating a complication of the structure of the ionosphere under certain conditions, and also the absence of a sufficient amount of experimental data, at present make it difficult to systematize the available material and to give a sufficiently complete physical explanation of the observed phenomena.

In some cases, during pulse sounding of the ionosphere, rapidly flickering reflections appear, with a duration of the order of 0.5 sec, and sometimes longer, separated by intervals of the order of 30 sec. At the same time there may be several such reflections. The height diagram, taken at a fixed frequency, in such cases has the appearance shown on oscillogram VII. The figure gives the results of observations at a frequency above the critical one. The effective heights of these reflections vary from several tens to several hundreds of kilometers. They are observed both by day and by night at different times of the year.

V

VI

VII

VIII

Irregular reflections of this type also appear in the form of a diffuse signal, stable in time, which apparently consists of a group of signals following one after another. Groups of reflections correspond, as a rule, to large delay times (10–20 milliseconds and more).

Direction finding of the sporadic signals described above shows that they arrive from all possible directions[^38]. From an analysis of their paths of arrival it follows that they are reflected predominantly in the \(E\) layer and in the region intermediate between the \(E\) and \(F\) layers. This type of reflection appears most often with powerful and, in particular, directed radiation from the transmitter. In these cases reflected signals are observed whose bearing at the receiving point corresponds to the direction of radiation of the transmitting device.

The type of reflections described above indicates the presence in the ionosphere of more or less random ionized clouds, from which irregular reflections occur. These clouds are most easily detected at the boundary of the skip zone, where there are no steady reflections; this also explains the circumstance that flickering reflections arrive from all possible directions, since reflections may then come from all sides. The frequency range in which reflections are most often observed corresponds to the short-wave range (20–30 m). Apparently, the ionized clouds have linear dimensions considerably greater than the wavelength, since individual random signals of this type preserve their form. Diffuse signals arise in this case as a result of the superposition of successive signals reflected from different parts of the cloud (from its edges), or from different clouds. More often it is not individual clouds that appear, but small groups of them. This can be seen from the fact that, with directed radiation, steady reflections are observed from the direction in which the ionosphere is, as it were, “illuminated”; the secondary light of this radiation is the reflected group of signals.

There are also experimental data[^52],[^53] indicating the frequent appearance of clouds both in the \(F\) layer and above it. Recently, with the aid of an apparatus making it possible to take height–frequency characteristics of the ionosphere every 10–30 sec, ionized clouds have been detected[^54] which move in the region of \(F\) with a velocity of 1–2 km/sec.

The reflections described are usually considered to be the result of scattering, although their physical nature does not fully correspond to what is commonly called, in optics, the scattering of light, when diffraction of an electromagnetic wave occurs at inhomogeneities with dimensions of the order of, or smaller than, the wavelength \(\lambda\)[^43]. Another type of reflection is observed, however, which apparently more closely fits what constitutes Rayleigh scattering[^55]. These have been studied still less. There is an indication that they appear more often at night.

The pattern shown in oscillograms VIII is observed. In case of

during vertical sounding of the ionosphere, a continuous stream of reflected signals appears, extending from several hundred to a thousand or more kilometers[^56]. The lower boundary of the effective heights of these reflections is almost independent of frequency and corresponds to the heights of the region \(F\). Over the entire frequency range the intensity of the reflections is almost constant, and the upper frequency limit, at which the reflections break off rather sharply, corresponds to \(10\)—\(12\) MHz. On the oscillograms \(a, b, c, d\), which were taken consecutively at intervals of 30 minutes, it can be seen how this type of reflection ceases during sunrise. Their lower frequency boundary gradually increases, and normal reflections from the \(F\) layer appear. On oscillogram \(c\) there still remains a narrow frequency range in which scattering is observed; later, however, it disappears completely.

The observed phenomenon may find its explanation in the fact that, in the region \(F\), in some cases there forms an accumulation of ionized clouds with linear dimensions smaller than, or of the order of, \(25\)—\(30\) m (corresponding to the upper frequency limit \(10\)—\(12\) MHz). This accumulation of clouds may have a considerable thickness. Scattering by these clouds, together with multiple reflections, leads to an apparent blurring of the signals. If this is Rayleigh scattering, then the ratio of the intensity \(J\) of the reflected signals to the intensity \(J_0\) of the incident signals must be proportional to

\[ \frac{J}{J_0} \sim \frac{1}{\lambda^4}(n-n_0)^2, \tag{5,1} \]

where \(n\) and \(n_0\) are, respectively, the refractive indices of the ionized clouds and of the medium surrounding them. Since in the ionosphere

\[ n \sim N\lambda^2, \]

we obtain

\[ \frac{J}{J_0} \sim (N-N_0)^2. \tag{5,2} \]

From (5,2) it is evident that the intensity of the scattered wave in this case does not depend on frequency, which is also observed in the experiment described. Scattering ceases at shorter waves, for which the linear dimensions of the clouds are comparable with or greater than the wavelength. Here regular reflections from individual clouds might occur; however, the reflection coefficient is very small if the ionization of the cloud only slightly exceeds the ionization of the surrounding medium, which is transparent for these frequencies.

The gradual recession of the lower frequency boundary of these reflections with sunrise may be explained by the fact that the accumulation of clouds was located at greater heights, so that at night, at small values of the ionization of the \(F_2\) layer, even low frequencies reached thesehiqizo

heights. With an increase in the ionization of the layer \(F_2\), reflections began in the lower part of the frequency range from the lower regions of the layer, initially partially and then completely covering this accumulation of ionized clouds, when the critical frequency assumes values of \(\sim 10\) MHz.

Another type of reflection observed in the \(F_2\) layer is represented in the height–frequency characteristics shown in oscillograms VI. These reflections are characterized by the fact that, in the region of the critical frequencies of the \(F_2\) layer, a strong blurring of the ordinary and extraordinary signals is observed. In some cases such a picture is obtained that the entire interval between the tails of the ordinary and extraordinary waves is filled with reflections. In the case shown in these oscillograms, the value \((f^x - f^0)\) between the edges or the middle of the tails of the characteristic corresponds to the value normal for the observation point,

\[ \sim \frac{f_H}{2}, \]

whereas the frequency difference \((f_2 - f_1)\) between the edges of the characteristic assumes all possible values in the interval \(f_H\). In some cases, blurring of the signal is observed not only in the region of the critical frequencies, but also in the lower part of the height–frequency characteristic, where splitting of the signal is not yet noticeable.

The available data show that effects of this kind occur more often at night in winter. The number of cases in which they are observed, relative to the total number of recorded characteristics, amounts in some months, according to preliminary data, to as much as 10%. The entire development of the process sometimes takes place over several hours.

Analysis of the results of measurements in which these effects are observed has made it possible to suggest that, in a number of cases, a fairly stable accumulation of ionized clouds arises in the region of \(F_2\), filling the layer almost throughout its entire thickness.^37 The degree of ionization of this accumulation of clouds varies with height, analogously to the ionization of the normal \(F_2\) layer, and reaches a maximum differing in magnitude from the maximum of the \(F_2\) layer, according to preliminary data, by 30–40%, and located at approximately the height of the normal \(F_2\) layer. This accumulation of ionized clouds, by analogy with \(E_{\text{spor}}\), has been called the sporadic \(F\) layer (\(F_{\text{spor}}\)).

Under such conditions the following occurs. On the one hand, there are reflections from the normal \(F_2\) layer, in which the wave propagates more or less without hindrance owing to the semitransparency of \(F_{\text{spor}}\). In the region of its critical frequencies, two tails are obtained on the height–frequency characteristic, corresponding to the ordinary \((f^0)\) and extraordinary \((f^x)\) waves. On the other hand, there are reflections from \(F_{\text{spor}}\), which, if its thickness is large and approximately coincides with the thickness of \(F\), may lead to a thickening or even a doubling of the lower part of the height–frequency characteristic and to the appearance of two additional tails on the characteristic,

corresponding to the critical frequencies of the extraordinary and ordinary waves \(F_{\text{spor}}\). In the case where the values of the critical frequencies \(F_{\text{spor}}\) differ little from the values of the critical frequencies of the normal layer \(F_2\), the signals become thickened—their apparent blurring—owing to the superposition of two ordinary and two extraordinary signals.

Characteristics of the type shown on oscillogram VI б may be obtained as a result of rapid changes in the maximum ionization \(F_{\text{spor}}\) owing to the motion of \(F_{\text{spor}}\), or when there is not one but several systems of such clouds. Oscillogram VI в gives the case of two cloud systems (6 tails).

г) “World” and “round-the-world” echo

This interesting radio phenomenon attracted the attention of investigators as long as 20 years ago. After the first report it was discussed in the literature for several years; but already

Fig. 23.

for about 10 years no discussion of it, nor any additional data about it, had been presented, despite the fact that the physical nature of this phenomenon has not been definitively clarified and its study is of interest.

In 1927 it was first indicated in the literature\(^{57}\) that, on short waves, echoes of telegraph signals had been observed with a delay of 3 sec. After this report, in 1928 joint observations\(^{58}\) were carried out at several points of the operation of a station specially emitting signals for the detection of these echoes. The experiments began in March; however, only in October were echoes with a delay of 3–15 sec first recorded. Subsequently, echoes were observed only on October 24. Some of these echoes (more than 100) were observed simultaneously at different points, and several of them at all points.

Later, various investigators observed (in 1930 and 1934) echoes with different delay times within the limits of 3–30 sec and more. The echoes observed in 1934 are presented in Fig. 23.\(^{59}\) From the figure it is seen that the delay time

echoes seem to have a maximum near 9–10 sec and 25–30 sec. It should be noted that in the observations of 1928 the greatest number of echoes were delayed by 8 sec; some accumulation of them was also observed after 25 sec.

In the communication cited above^57 it was suggested that the delayed echo consists of signals which penetrate through the ionosphere and are reflected near a toroidal surface formed around the Earth by the stream of electrons emitted by the Sun. According to this theory, developed as early as 1904, streams of particles emitted by the Sun move, under the influence of the Earth’s magnetic field, along complex orbits, forming a toroidal surface with an axis coinciding with the Earth’s magnetic axis (Fig. 24). This surface is at very great distances from the Earth.

Fig. 24.

The surface of this torus consists of electrons, while in its inner cavity there are no particles of any kind. The electrons, moving around this surface, enter the poles, where they produce auroras. It should be noted that this theory has been confirmed in many of its details as applied to auroras.

Alongside this hypothesis, another explanation of these echoes was also discussed. It was suggested that under certain conditions the signal is reflected from ionized regions having a refractive-index value close to zero, which corresponds to small (approaching zero) values of the group velocity. Propagation in such regions leads to a long delay time. Since propagation in such a medium for several seconds would cause very strong attenuation of the wave, it was supposed that these ionized regions are formed above the ionospheric layer $F_2$, where the atmospheric density is very small and the number of collisions reaches only a few per second in 1 cm^3.

At present there is still not a sufficient quantity of data that could definitively indicate which of these hypotheses is the more correct.

It should be noted, however, that the assumption of the occurrence of ionization regions with $n \sim 0$ for a given frequency is in itself unlikely. Moreover, prolonged propagation in such

in the medium will lead to a considerable spreading of the signal (see above), which will make its detection very difficult. Already from a simple estimate it is evident that under these conditions the signal spreads out strongly.

For the group velocity in the ionosphere, if \(H_0=0\), one obtains the expression

\[ u=c\cdot n=c\sqrt{1-\frac{a}{f^2}}. \tag{5,3} \]

Suppose that a signal of duration \(\Delta t=10^{-3}\) sec, and consequently \(\Delta f\sim 10^3\), propagates in a medium where \(n\sim 10^{-2}\). From (5,3) we obtain

\[ \frac{du}{df}=\frac{cu}{nf^3} \tag{5,4} \]

and since \(n\approx 10^{-2}\) and \(\dfrac{a}{f^2}\sim 1\), then

\[ u_{f+\Delta f}=u\left(1+\frac{1}{n^2}\frac{\Delta f}{f}\right), \tag{5,5} \]

whence it follows that at \(f=10^7\) the group velocities of the boundaries of the signal \((\Delta f=10^3)\) differ by a factor of two, which leads, roughly speaking, to the fact that its duration \(\Delta t\) must be of the order of the time of its delay (twice smaller). Experiments, however, show that the echo is a quite clear, undistorted signal.

Along with the phenomenon of the “world echo” indicated above, in 1926 “round-the-world echoes” were first described\(^{60}\), i.e. signals that had gone around the terrestrial globe. Subsequently, both single and multiple circumnavigations of echo signals around the Earth were observed. According to measurement data, the time for circumnavigation of the Earth is approximately \(\dfrac{1}{7}\) sec \((0.138\) sec), i.e. it exceeds by several percent the time required for propagation over a distance equal to the perimeter of the terrestrial globe.

In the literature two considerations have been expressed concerning the trajectories of these echoes. One of them is based on the fact that the ray trajectory is circular—that is, that the signal moves along a circumference whose radius exceeds the radius of the Earth by the height of the ionosphere\(^{61}\).

According to another supposition, the ray trajectory is zigzag-shaped—that is, it corresponds to a path with multiple reflections from the Earth and the ionosphere\(^{62}\). In one of the recent works\(^{63}\) the phenomenon of the round-the-world echo is again subjected to analysis. In this work the author suggests that the daytime echo follows a trajectory which he calls “ricocheting,” i.e. that the signal propagates as if along a curved waveguide channel formed in the ionosphere.

d) Nonlinear Effects

It is also useful to dwell on a very interesting phenomenon observed in the ionosphere, the study of which may apparently help to elucidate important details of the microprocesses occurring in the ionosphere and to refine some of its constants.

This phenomenon, called in the literature the Luxembourg–Gorky effect,^{64,66} consists in the fact that in some cases an interaction is observed in the ionosphere between radio waves of different wavelengths propagating through it. The theoretical analysis of this effect cannot yet be considered fully developed. Moreover, from the experimental side as well it has not been sufficiently investigated.

The effect described manifests itself in the fact that, when receiving the transmission of a radio station operating at one frequency \((\omega_1)\), one hears the transmission of a radio station operating at another frequency \((\omega_2)\), substantially different from the frequency \(\omega_1\). This listening often occurs during breaks in the modulation of the radio station of frequency \(\omega_1\), when it is emitting only the carrier frequency.

Soon after the first report of this phenomenon, it was suggested^{65} that the cause of the observed effect might be the circumstance that the radio station of frequency \(\omega_2\), especially if it is powerful, affects the velocity of the thermal motion of electrons in the ionosphere.

Under these conditions the number of collisions of electrons with molecules \(\nu\) becomes a function of the electric field of this radio station and, thus, the quantity \(\nu\) may undergo changes in step with the transmission of the powerful station. If the reflection of another radio wave \((\omega_1)\) occurs in that region of the ionosphere which is subject to the influence of the powerful radio station \((\omega_2)\), then the modulation \((\omega_2)\) is imposed on the carrier frequency of this wave \((\omega_1)\). The depth of cross-modulation obtained in this way is small; therefore the radio station of frequency \(\omega_2\) is often heard only during breaks in the modulation of the station being received \((\omega_1)\).

In the first calculations^{65} it was established that the depth of cross-modulation depends on \(\nu\), so that the corresponding measurements can be used to determine its value. It was also pointed out that similar nonlinear effects apparently also manifest themselves in that strong atmospheric disturbances occurring in the regions of reflection of the received radio wave are superimposed on its modulation and thereby distort it.

In addition, in these works it was shown that in the ionosphere there must occur a peculiar resonant interaction of radio waves, consisting in the fact that if the wavelength is close to the gyroscopic frequency, then its action on the ionosphere increases so much that even at relatively low radiated powers—

radiation it will cause a noticeable cross-modulation of radio waves reflected from the region of the ionosphere subjected to its influence. Observations of transmissions from not particularly powerful radio stations operating at frequencies close to the gyrofrequency confirmed this assumption.

However, the theoretical treatment given in the works cited was not based on a detailed kinetic analysis of processes in an ionized gas, which made it possible to obtain formulas only up to numerical coefficients. Moreover, these works did not take into account collisions of electrons with ions, which play an essential role in the processes occurring in the ionosphere.

A more exact theoretical analysis of this question, using the kinetic equation for electrons and taking into account their collisions with ions and molecules, has recently been carried out[^67]. In this work formulas were obtained which make it possible, in certain cases, to discuss the experimental data in considerable detail and to form an idea of various details of the microprocesses occurring in the ionosphere, in particular of the role of collisions of electrons with molecules or ions.

It should be noted that, from the experimental side, the Luxembourg–Gorky effect, according to the literature, has been studied very little, and no details of it are known. It may be supposed that the large network of powerful radio stations which has appeared in recent years will lead to a more complete investigation of this interesting phenomenon from various points of view.

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